Bun is apparently targeting September 7th for a 1.0 release. I also have concerns about delivering a stable api using a language that doesn't have one yet.
Tbh I've had the same issue delivering my own 1.x software with 0.x dependencies and my argument has always been that it's my problem, not the client's, so "do not worry". Being realistic, no sw offers any guarantees and Zig's 1.0 stability is more of a "contract" than a "guarantee".